Whatever the choice of language, there will only be a countableinfinity of possible texts, since these can be listed in size order,and among texts of the same size, in alphabetical order. {{Here is asimple example:0100011011000...}} This has the devastating consequence that there are only adenumerable infinitely of such "accessible" reals, and therefore theset of accessible reals has measure zero. So, in Borel's view, most reals, with probability one, aremathematical fantasies, because there is no way to specify themuniquely. Most reals are inaccessible to us, and will never, ever, bepicked out as individuals using /any/ conceivable mathematical tool,because whatever these tools may be they could always be explained inFrench, and therefore can only "individualize" a countable infinity ofreals, a set of reals of measure zero, an infinitesimal subset of theset of all possible {{interesting question: what are /possible/properties of /possible/}} reals. Pick a real at random, and the probability is zero that it'saccessible - the probability is zero that it will ever be accessibleto us as an individual mathematical object. {{How can we pick? Bypicking it, a real number would be finitely defined already. Thatmeans an undefined real number can never be picked mathematically. Andwith finger or beak nobody could succed.}}[Gregory Chaitin: "How real are real numbers?" (2004)]http://arxiv.org/abs/math.HO/0411418

The enumeration of all rational numbers is tantamount to an infinitesum of units. One gets the divergent sequence of all finite cardinalnumbers and maintains that a limit exists. That is a mistake. The factthat we can count up to every number does not imply that we can countall numbers. After every finite cardinal number there are infinitelmany but not after all. In a similar way it is impossible to sum allterms of the series SUM 1/2^n. But contrary to a diverging sequence,the sequence of partial sums of this series deviates from 1 less andless. Therefore 1 can be called the limit.